In the decades after the second world war, the celebrated architect Aldo van Eyck designed more than 700 playgrounds in Amsterdam, filling bomb sites with dazzling constellations of tumbling bars, leapfrog posts and climbing domes. His idea was that by providing children with a range of elemental forms and open-ended structures – rather than swings, roundabouts and other playground staples – their creativity would be stimulated and they would invent new games.

These “tools for the imagination”, as he called his kit of sandpits, frames and posts, became a familiar part of Amsterdam’s streetscape, a connected galaxy of playtime fragments that spread across the city, from public spaces and even to roadside verges, never fenced off. It was a vision of play without walls, the protected domain of the child thrown open and spilling over into the rest of the city.

Van Eyck’s pioneering work is one of the highlights of Play Well, a new exhibition at the Wellcome Collection in London. This fun, candy-coloured romp through the transformative power of play is nothing if not wide-ranging: it explores the work of Friedrich Fröbel, inventor the kindergarten; the role of ad-hoc “play labs” set up by humanitarian organisations in refugee camps; and the rarefied realm of adult larping, or live-action role-playing, for those who wish it was Halloween every day – from battle re-enactors to bonnet-wearing Jane Austen superfans.

For Van Eyck, the playground was the place where the child could be “lord of the city”. As well as extending children’s sense of urban ownership, his fence-free compounds encouraged a more relaxed attitude to risk. Some were even located on the central reservations of roads – forcing children and cars to be more aware of each other, making the city feel like a more equal, accessible place.

Van Eyck’s work was lauded by architects and psychologists, but the idea of kids freed from fences set alarm bells ringing among the health and safety lobby in more conservative parts of the world, and his ideas were rarely repeated elsewhere.

This sense of owning the streets is also captured in the postwar photography of Bert Hardy, Nigel Henderson and Shirley Baker, displayed alongside Van Eyck’s designs. Their striking black-and-white images depict a bygone era of kids playing outside, using whatever scraps they could find, very much lords of the city with their catapults and toy guns at the ready. There’s an air of anarchy, one that children’s rights activist Lady Marjory Allen encouraged in her promotion of junk playgrounds around the same time. These were later rebranded as adventure playgrounds to make the idea more palatable to nervous parents and local authorities.

“Children are needlessly overprotected by adults,” she says in a film from the 1970s. “They’re never given any freedom to explore and experiment and find out what the world is all about.” The camera then cuts to kids rampaging around one of her wild playgrounds, full of scavenged junk and firepits. “Here, they can play with very dangerous tools,” she says proudly, in a cut-glass accent. “They can create their own houses, their own climbing frames. They can take really dangerous risks and overcome them. And above all, it’s a place where they can meet their friends, where they can make new friends, in a very free and permissive atmosphere.”

Allen had been inspired by a trip to Denmark in 1945, where she saw the work of architect Carl Theodor Sørensen. His skrammellegepladsen, or junk playgrounds, were visions of creative chaos, made mostly by children themselves. She helped set up 17 trial junk playgrounds in the UK, equipped with makeshift treehouses, walkways, nets, ropes and rubber tyres. The very first, at Lollard Street in London’s Kennington, is still going strong.

“Ideas about play haven’t changed much since then,” says Nicola Butler, chair of Play England, who co-authored the charity’s Design for Play guidance in 2008 – and then discovered that Allen had written a pamphlet of the same name in 1962, outlining almost identical principles. “The more objects that children can actually manipulate themselves, the more enjoyment they will get out of a playground.”

Butler was concerned that playgrounds had become increasingly risk-averse, as a result of the claims culture of the 1990s and 2000s. “Climbing frames were getting smaller, roundabouts and swings were going more slowly and being designed with more restrictive movement. When playgrounds become really safe and boring, kids climb on top of bus shelters instead because it’s more fun.”

This pro-risk view is shared by landscape architect Jennette Emery-Wallis, designer of the award-winning Diana Memorial Playground in Kensington Gardens and the equally acclaimed Tumbling Bay in Olympic Park, designed with Erect Architecture. The former is a piratical Peter Pan fantasy, featuring a great galleon marooned in a sandy sea, with rigging to climb, a crow’s nest to look out from, and a hold to explore, surrounded by a magical landscape dotted with teepees, Wendy houses and dammable water channels.

“Until recently, many playgrounds were just the result of going to a supplier and asking for a KFC,” says Emery-Wallis, referring not to a finger-lickin’ bucket of fried chicken, but a “kit, fence and carpet”, the go-to trinity for the lazy playground procurer. “With the Diana project, we tried to break the mould, designing a place full of character and texture that kids would want to explore, giving them autonomy to make it their own.” Rather than a playground dotted with equipment, it is conceived as an entire landscaped world, with specific zones, each with their own characters and themes from the Peter Pan story.

Like Butler, she says the key is allowing children to manipulate their environment, the two crucial ingredients being sand and water. At Tumbling Bay, visitors are drawn through a hazel copse and birch woodland, encouraging them to make their own dens, next to an elaborate sand and water area where they can become mini-engineers. There’s also a more adventurous woodland with treetop pods and rope bridges. The lack of health and safety paraphernalia is refreshing, with sheer drops, rope swings and four-metre-high fireman’s poles.

It’s all designed, says Emery-Wallis, “for proper risk-taking and getting dirt under your fingernails”. It might look more expensive than your average council KFC, but 80% of the materials were scavenged from Olympic leftovers. As for the increased dangers, she says Play England has a rigorous assessment system that measures the risk against benefit. “You’ve got to have a sense of risk and excitement,” she says, “otherwise we’re cosseting our kids in a bubble-wrapped world.”

In Berlin, the adventurous spirit has been taken to the extreme in the form of Kolle 37, a mini-metropolis of tree forts, walkways and dens, built entirely by children, with a little grownup supervision. Along with plentiful supplies of hammers, saws, nails and wood, it has pottery kilns, a blacksmith’s forge and a bike rental shop where older kids can work – making it not only the city’s most fun playground, but its most industrious.

As attitudes to risk relax, some of Van Eyck’s ideas are gaining ground once again. Emery-Wallis’s practice, Land Use Consultants (LUC), is working with developers and councils to integrate fence-free play areas into their housing schemes, attempting to expand play out into the city, in accordance with the Dutchman’s vision. “I hope we end up full circle,” she says, “with even wilder, riskier play opportunities everywhere – and not just for children aged two to 10. We should be thinking about multigenerational play, tackling loneliness, getting everyone more active outdoors.”

It could be the gateway to a more equitable city, one that works for everyone. As Van Eyck put it: “If we create a playground well, we create a world in which man rediscovers what is essential – and the city rediscovers the child.”

• Play Well is at the Wellcome Collection, London, until 8 March.